Improved Small-x Evolution with Running Coupling Effects DIS03, 24 April ‘03 CERN G. Altarelli Based on G.A., R. Ball, S.Forte hep-ph/ (NPB 575,313) hep-ph/ (lectures) hep-ph/ (NPB 599,383) hep-ph/ More specifically on hep-ph/ (NPB 621,359) and on CERN-TH/ now being completed
G. Altarelli Our goal is to construct a relatively simple, closed form, improved anom. dim. I ( ,N) I ( ,N) should reduce to pert. results at large x contain BFKL corr’s at small x include running coupling effects be sufficiently simple to be included in fitting codes closely follow the trend of the data and of course
G. Altarelli Moments For each moment: singlet eig.vector with largest anomalous dim. eig.value Singlet quark Mellin transf. (MT) Inverse MT ( >0) t-evolution eq.n : anom. dim known Pert. Th.: LONLO
G. Altarelli In principle the BFKL approach provides a tool to control ( /N) n corrections to (N, ), that is 1/x( log1/x) n to splitting functions. Define t- Mellin transf.: With inverse: -evolution eq.n (BFKL): with known Bad behaviour Bad convergence
G. Altarelli M s =0.2 0 + 2 1 0 The minimum value of 0 at M=1/2 is the Lipatov intercept: 0 = 0 (1/2)= c 0 = 4n c / log2~2.65 ~0.5 It corresponds to (for x->0): xP(x)~x - 0 Too hard not supported by data
G. Altarelli In the region of t and x where both are approximately valid, the "duality" relation holds: Note: is leading twist while is all twist. Still the two perturbative exp.ns are related and can improve each other. Non perturbative terms in correspond to power or exp. suppressed terms in
G. Altarelli For example at 1-loop: 0 ( ( ,N))=N/ 0 improves by adding a series of terms in /N) n 0 -> Similarly it is very important to improve by using 1l. Near M=0 0 ~ 1/M, 1 ~ -1/M 2 Duality + momentum cons. ( ( ,N=1)=0) { Double Leading Expansion
G. Altarelli The DL expansion is OK near M=0. But is still bad near M=1 (no analogue of mom. cons.). One can try to use symmetry of underlying BFKL kernel (broken by scale unbalance in DIS) Ciafaloni,, Salam +Colferai, Camici or introduce some additional idea to treat the central region GA, Ball, Forte
G. Altarelli The implementation of running coupling in BFKL is not simple. In M-space -> operator In leading approximation: A perturbative expansion in 0 leads to validity of duality with modified and But expansion fails near M=1/2: 0 '(1/2)=0
G. Altarelli DL-LO DL-NLO Res At M=1/2 0 has a minimum and 1 is singular (and also ss ). We shall see it is just an artifact of pert. exp.
G. Altarelli By taking a second MT the equation can be written as [F(M) is a boundary condition] It can be solved iteratively or in closed form: H(N,M) is a homogeneous eq. sol. that vanishes faster than all pert. terms and can be dropped.
G. Altarelli The following properties can be proven: From G(N,M) we can obtain G(N,t) and evaluate it by saddle point expansion. This G(N,t) satisfies duality (in terms of modified and according to the perturbative results singular at '(1/2)=0) and factorisation(no t-dep. from the boundary condition). From G(N,M) we can obtain G( ,M). This presents unphysical oscillations when >0 for all M. These problems can be studied by using the Airy approximation: The asymptotics is fixed by the behaviour of near the minimum, where a quadratic form is taken (here c is free and k as in : Lipatov; Collins,Kwiecinski;Thorne; Ciafaloni, Taiuti,Mueller G.A., R. Ball, S.Forte hep-ph/ (NPB 621,359)
G. Altarelli The explicit solution is where Ai''(z)-zAi(z) = 0 Ai(0) = 3 -2/3 (2/3) Ai(z) z z 0 ~-2.338
G. Altarelli From one obtains G(x,t) by inv. MT The asymptotics is dominated by the saddle condition: For c>0 at not too large this is satisfied at large N. When increases N gets smaller. Then oscillations start, d/dN changes sign and the real saddle is lost. G( ,t) starts oscillating, in agreement with the general analysis.
G. Altarelli Ai[z(N)] N c= -1 =0.2 N Ai[z(N)] c= +1 =0.2
G. Altarelli The dual anom. dim. A is given by N large N0N0 z0z c=1 s =0.2
G. Altarelli The splitting function is completely free of oscillations at all x!! 2-loop c= The oscillations get factorised into the initial condition
G. Altarelli The Airy result is free of the perturbative 0 singularities. At NLL order we can add the full A and subtract its large N limit: The last term cancels the sing. of ss (N= c corresponds to M=1/2) 0 -> s, 1 -> ss
G. Altarelli The effect of running on in the Airy model Is a softer small-x behaviour c=1 k=-2n c / ’’ (1/2) = cc N0N0 xP ~ x - xP ~ x - N 0
G. Altarelli c=2, 1.5, 1, 0.5 =c s (Q 2 ) N 0 (Q 2 ) As an effect of running, the small-x asymptotics is much softened: xP ~ x - xP ~ x - N 0
G. Altarelli The goal of our ongoing work is to use these results to construct a relatively simple, closed form, improved anom. dim. I ( ,N) I ( ,N) should reduce to pert. result at large x contain BFKL corr’s at small x include running coupling effects (Airy) be sufficiently simple to be included in fitting codes closely follow the trend of the data and of course here I will describe the LO approximation G.A., R. Ball, S.Forte, CERN-TH/
G. Altarelli Improved anomalous dimension 1st iteration: optimal use of 1l (N) and 0 (M) Properties: Pert. Limit ->o, N fixed o( 2 ) Limit ->o, /N fixed +o( ) Pole in 1/N Cut with branch in c 0 the Airy addition cancels the cut and replaces it with a pole at N=N 0
G. Altarelli II GLAP LO&NLO DL s =0.2 DL ( , N )=
G. Altarelli xP 1/x GLAP LO&NLO II DL s =0.2 Here we have the same plot for the corresponding splitting functions. Note: for s =0.2 the pole in GLAP is ~0.191/N while the pole in I is ~0.014/(N-N 0 ) (only visible at very small x)
G. Altarelli Conclusion BFKL with running coupling is fully compatible with RGE t evolution, factorisation and duality. Using the Airy solution we have seen that the splitting functions are completely free of unphysical oscillations (can be factorised in the initial condition at t 0 ). The Airy solution can be used to resum the perturbative singularities in the 0 expansion We use these results to construct an improved an. dim. that reduces to the pert. result at large x and incorporates BFKL with running coupling effects at small x. Properly introducing running coupling effects in the LO softens the asymptotic small x behaviour as indicated by the data.